I've been studying probability theory in one form or another for most of my career, and I have quite a bit of expertise in stochastic processes, statistical methods and information theory.
However, I've tended never to really worry about measure theory, preferring instead to work only with discrete probability distributions, only generalising to continuous variables in the specific, limited cases where I've found a need to do so.
I'd like to learn a bit more about the measure-theoretic foundations of my field, but the resources I've found so far have been rather heavy going. For some reason the notion of probability tends to be entirely absent from the motivation, and we're asked to consider sets of sets with certain abstract properties and prove theorems about them, without really knowing where we're going or why we need to do it. I can follow this sort of thing if I have to, but I can't help feeling there should be an easier way, given my own particular background.
So I'm wondering if there exists a good gentle introduction to measure theory, written for people who work with probability already, rather than as an introduction to probability for people used to more abstract topics. Instead of starting with "a measurable space is a set equipped with a $\sigma-$algebra", I'm looking for something that would start with "these are the problems that occur if you try to do probability on infinite sets in a naïve way, and this is why defining measurable spaces in this particular way helps us to resolve them."
I note that there are multiple previous questions asking for introductions to measure theory. However, to my knowledge none of them are coming from this particular perspective, of wanting an introduction aimed at people who already do calculations in probability theory and want to understand the foundations.
I recommend Billingsley's book "Probability and measure". The main focus is probability theory and the measure theory is developed along the way.